KöMaL 2013 A594

The convex quadrilateral ABCD has an inscribed circle which is tangent to the sides $AB$ and $BC$ at $E$ and $F$, respectively. The diagonal $AC$ meets the inscribed circle at $P$ and $Q$; the point $Q$ is located between $A$ and $P$. Show that the lines $BD$, $EP$ and $FQ$ are concurrent.